1. Technical Field of the Invention
The invention relates generally to communication systems; and, more particularly, it relates to constellation rotational errors that may be experienced in such communication systems dues to effects such as carrier frequency offset.
2. Description of Related Art
Data communication systems have been under continual development for many years. One very effective type of coding employed within many communication systems is that of Trellis Coded Modulation (TCM). When coding symbols using TCM, there are a variety of types of constellation to which the symbols may be mapped. For example, one type of symbol includes 8 PSK (Phase Shift Keying) mapped symbols. Each of the symbols in this constellation type typically have a common magnitude and are separated in phase around the I,Q (In-phase, Quadrature) plane's origin. One difficulty that can arise when employing an 8 PSK modulation (that includes the 8 PSK shaped constellation and a particular mapping of the constellation points in that constellation) is when the axes of the I,Q plane are rotated by some offset. A number of sources may cause this offset between the 8 PSK constellation by which 8 PSK symbols are encoded and an 8 PSK constellation at which the 8 PSK symbols are decoded. One potential source of such rotational offset may be a Carrier Frequency Offset (CFO) between the local oscillators employed at the transmitter end of a communication channel and at the receiver end of the communication channel.
If this rotational offset does occur, then the mapping of the symbols may be performed incorrectly. For example, the mapping of the 8 PSK symbols to the appropriate constellation points within the 8 PSK constellation may be performed erroneously.
There have been some attempts in the art to try to generate a code that is resilient and operable to code 8 PSK symbols such that they are rotationally invariant; in other words, some attempts have been made to try to ensure that the information encoded using TCM coding and an 8 PSK shaped constellation may be able to accommodate a rotational offset and still be able to communicate data effectively from one place (e.g., a transmitter) to another (e.g., a receiver). However, attempts to do this have, so far, been met with serious problems that have yet to be overcome.
More specifically, up to now, any constructed code that is operable to encode 8 PSK symbols using TCM without parallel transitions is either not rationally invariant at all or else has a catastrophic operation as described in the following references: [1] A. Fung and P. McLane, “Phase jitter sensitivity of rotationally invariant and 16 point trellis codes,” IEEE Proceedings-I, Vol. 138, No. 4, August 1991, pp. 247–255, [2] E. Biglieri, D. Divsalar, P. J. McLane and M. K. Simon, Introduction to trellis-coded modulation with application, Macmillan Publishing Co., New York, 1991, and [3] W. Liu and S. G. Wilson, “Rotationally-invariant concatenated (turbo) TCM codes,” Signals, Systems and Computers, 1999. Conference Record of the Thirty-Third Asilomar Conference on, Vol. 1, 1999, pp. 32–36.
While there has clearly been a significant amount of effort in the field of communications to try to address this rotational problem within 8 PSK constellations, no solution has yet to present an effective solution to overcome this problem. Moreover, in the context of TCM coding that is performed using no parallel transitions, there has yet to be an effective solution presented in the art to overcome this problem.